3/12/2010
First of all, thanks to Smita who helped to teach today's class.
Wow! Today our 2nd grade math class solved the very difficult problem. I am still amazed that they did it.
There are 2 empty jars. The first jar can be filled with 5 ounces of water and the second jar with 3 ounces.
How do you fill the largest jar with 1 ounce of water? You can fill a jar or empty it, if necessary, and use any amount of water.
Before going to the class I asked several adults to solve the problem. Some adults found a solution within several minutes others spent hours. Before reading how to solve the problem try yourself. When preparing for the class I did not realize that the problem can be solved in more than one way.
We divided the class in two groups, so Smita worked with one and I with another. I brought 2 pairs of glasses. Each group were given 2 glasses: one glass represented the largest jar (5 oz) and another the smallest one (3 oz).
Kids started with running to the water fountain, filling glasses with water and pouring water from one glass to another. Some kids suggested to fill in the largest glass with 5 oz, pour water into the smallest glass leaving 2 oz in the largest one. However they realized that they could not get precisely 1 oz. Kids tried several ways, but nothing worked. They thought hard for about 10 minutes. Then we asked them to stop and we switched to other problems. We told them that this will be their homework...
Because kids finished all the problems that we prepared for them fast we had some time to work at 'jar' problem again.
Smita guided her group through the thinking process. I believe that their solution was the following.
Kids made 5 oz glass full . Then they poured 3 oz from the largest into the smallest glass. So they had 2 oz left in the largest glass. Because kids could see through the glass they took a marker and marked the level for 2 oz on the glass. Then they emptied the largest glass, poured 3 oz from the smallest to the largest. Afterward kids poured water from the largest into smallest glass up to the mark they put earlier. This way they ensured that they had only 1 oz in the smallest glass. Then the largest glass was emptied and they poured 1 oz from 3 oz glass into 5 oz glass. The problem is solved!
My group solved the problem by accident :) One kid suggested to make 3 oz glass full, pour water from that glass into 5 oz glass, and then repeat it one more time. He did not realize that he was so close to solving the problem. Then one girl said: Stop! If we put 3 oz of water from the smallest glass into the largest and repeat the same thing up to the point until 5 oz glass is full, then we would have exactly 1 oz in the smallest glass and we can put that 1 oz into the largest one afterward. The solution is found!
After the class one boy came to me and said: It was awesome!
Friday, March 12, 2010
What is the weight of the apple?
3/12/2010
I knew that 2nd graders study fractions this month, so I decided to give them 'fraction' problem.
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
To determine the weight of 1/5, kids drew an apple and divided it into 5 parts. However it proved to be a very complicated task for them.
Then we started with more simple problem. Let's say the weight is not 400, but 100 grams. What is the weight of 1/2? Kids replied: 50 grams. Correct! What is the weight of 1/4? 25 grams, kids replied. Correct! What is the weight of 3/4? 75. Correct!
Now back to the original problem, the weight is 400 grams. What is the weight of 1/2? 200 grams. Correct! What is the weight of 1/4? 100 grams. Correct! What is the weight of 1/5? This was still difficult for kids.
I drew a circle representing an apple. Divided circle with 5 equal parts. Each part is 1/5. I marked each part with ?. We have ? + ? + ? + ? + ? = 400 grams => 5 x ? = 400 grams. What number do you need to take 5 times to get 400? Kids replied: 80. Correct! What is 2/5 then? Kids replied: 160. Correct! What is 3/5? 240. Correct!
I knew that 2nd graders study fractions this month, so I decided to give them 'fraction' problem.
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
To determine the weight of 1/5, kids drew an apple and divided it into 5 parts. However it proved to be a very complicated task for them.
Then we started with more simple problem. Let's say the weight is not 400, but 100 grams. What is the weight of 1/2? Kids replied: 50 grams. Correct! What is the weight of 1/4? 25 grams, kids replied. Correct! What is the weight of 3/4? 75. Correct!
Now back to the original problem, the weight is 400 grams. What is the weight of 1/2? 200 grams. Correct! What is the weight of 1/4? 100 grams. Correct! What is the weight of 1/5? This was still difficult for kids.
I drew a circle representing an apple. Divided circle with 5 equal parts. Each part is 1/5. I marked each part with ?. We have ? + ? + ? + ? + ? = 400 grams => 5 x ? = 400 grams. What number do you need to take 5 times to get 400? Kids replied: 80. Correct! What is 2/5 then? Kids replied: 160. Correct! What is 3/5? 240. Correct!
What is the price of the book?
3/12/2010
This is another fraction problem we did today.
We know that 1/5 of the book's price is $2. What is the price of the book?
Kids solved the problem within several seconds. The answer is 10. Too easy!
This is another fraction problem we did today.
We know that 1/5 of the book's price is $2. What is the price of the book?
Kids solved the problem within several seconds. The answer is 10. Too easy!
How to get from Boston to New York?
3/12/2010
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
I asked kids to draw 3 points on the white board representing Boston, Hartford, and New York City. They connected Boston -Hartford and Hartford - New York. Then to help them visualizing they drew bus and train next to Boston - Hartford line and bus, train, and plane next to Hartford - New York line. Then they started counting all combinations: bus - bus, bus - train, etc. One girl wrote 'bus' and 'train' on the left side of her white board and 'bus', 'train', and 'plane' on the right side. Then she connected all possible combinations.
The answer is 6.
Afterward I showed to them that instead of just counting all combinations one by one they could just multiple 2 by 3, where 2 is how many different ways to get from Boston to Hartford and 3 is how many ways to get from Hartford to New York.
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
I asked kids to draw 3 points on the white board representing Boston, Hartford, and New York City. They connected Boston -Hartford and Hartford - New York. Then to help them visualizing they drew bus and train next to Boston - Hartford line and bus, train, and plane next to Hartford - New York line. Then they started counting all combinations: bus - bus, bus - train, etc. One girl wrote 'bus' and 'train' on the left side of her white board and 'bus', 'train', and 'plane' on the right side. Then she connected all possible combinations.
The answer is 6.
Afterward I showed to them that instead of just counting all combinations one by one they could just multiple 2 by 3, where 2 is how many different ways to get from Boston to Hartford and 3 is how many ways to get from Hartford to New York.
Wednesday, March 10, 2010
Proposal for Math Lesson on March 12
Here is the proposed problem for next Friday.
How to fill a jar with water?
There are 2 empty jars. The first jar can be filled with 5 gallons of water
and the second jar with 3 gallons.
How do you fill the largest jar with 1 gallon of water?
What is the weight of the apple?
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
What is the price of the book?
We know that 1/5 of the book's price is $2. What is the price of the book?
How to get from Boston to New York?
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
How to fill a jar with water?
There are 2 empty jars. The first jar can be filled with 5 gallons of water
and the second jar with 3 gallons.
How do you fill the largest jar with 1 gallon of water?
What is the weight of the apple?
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
What is the price of the book?
We know that 1/5 of the book's price is $2. What is the price of the book?
How to get from Boston to New York?
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
Friday, March 5, 2010
How many games teams play in soccer tournament?
3/5/2010
This problem is very similar to 'handshakes' problem.
I had 2 groups of children. First group had 3 kids and the second had 4 kids. Each child represented a soccer team. All those teams are playing in soccer tournament. In first group there are 3 teams and in the second there are 4. Teams play against another 3 games. How many games played in each group?
To solve this I reminded children about 'handshake' problem. Each handshake represents a game.
Kids did handshakes and calculated total number of games. Kids in both groups calculated number of games correctly by just counting total number of handshakes among each other.
Then I asked them to step back. I reminded them how we could solve similar problems by applying arithmetic progression: 3 handshakes between 2 kids, 6 if add third kid, 9 when fourth kid is added. So we have 3+6+9=18.
After that I asked: What if we add 5th team? We would have 3+6+9+12=30.
I reminded them how we can calculate the sum of arithmetic progression. Add the first and the last numbers (3+12=15) and add the second and second from the last numbers(6+9=15). Then add those results: 15+15=30.
This problem is very similar to 'handshakes' problem.
I had 2 groups of children. First group had 3 kids and the second had 4 kids. Each child represented a soccer team. All those teams are playing in soccer tournament. In first group there are 3 teams and in the second there are 4. Teams play against another 3 games. How many games played in each group?
To solve this I reminded children about 'handshake' problem. Each handshake represents a game.
Kids did handshakes and calculated total number of games. Kids in both groups calculated number of games correctly by just counting total number of handshakes among each other.
Then I asked them to step back. I reminded them how we could solve similar problems by applying arithmetic progression: 3 handshakes between 2 kids, 6 if add third kid, 9 when fourth kid is added. So we have 3+6+9=18.
After that I asked: What if we add 5th team? We would have 3+6+9+12=30.
I reminded them how we can calculate the sum of arithmetic progression. Add the first and the last numbers (3+12=15) and add the second and second from the last numbers(6+9=15). Then add those results: 15+15=30.
How many cats are on the farm?
3/5/2010
Today we had an interesting problem to solve and kids had a lot of fun.
I divided math group into two subgroups and asked each subgroup to solve the problem independently.
Here is the problem.
There are cats and chickens on the farm. A cat has 5 claws on one leg and a chicken has 4 claws on one leg. There are 104 claws all together. How many cats are on the farm?
One group gave me an answer right away: 5 cats and 1 chicken with only one leg :)
Because the answer was wrong kids continued solving it.
In a little while one group gave me the correct answer: 4 cats, but when I asked them how many chickens they told me 24. I guess all those 24 chickens had 1 leg and 1 claw :) Kids understood what they did wrong and gave me the correct answer: 3 chickens. The second group solved that problem just after the first one.
Then I told children that there is another solution. After a while both groups gave me the correct answer: 2 cats and 8 chicken.
After that I told them that this problem can have another answer. They could not solve it for a while and then I gave them a hint that number of cats can be any including 0,1,7, etc. They tried 0 and it worked. 0 cats and 13 chicken.
Today we had an interesting problem to solve and kids had a lot of fun.
I divided math group into two subgroups and asked each subgroup to solve the problem independently.
Here is the problem.
There are cats and chickens on the farm. A cat has 5 claws on one leg and a chicken has 4 claws on one leg. There are 104 claws all together. How many cats are on the farm?
One group gave me an answer right away: 5 cats and 1 chicken with only one leg :)
Because the answer was wrong kids continued solving it.
In a little while one group gave me the correct answer: 4 cats, but when I asked them how many chickens they told me 24. I guess all those 24 chickens had 1 leg and 1 claw :) Kids understood what they did wrong and gave me the correct answer: 3 chickens. The second group solved that problem just after the first one.
Then I told children that there is another solution. After a while both groups gave me the correct answer: 2 cats and 8 chicken.
After that I told them that this problem can have another answer. They could not solve it for a while and then I gave them a hint that number of cats can be any including 0,1,7, etc. They tried 0 and it worked. 0 cats and 13 chicken.
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